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Chapter 14: Q. 9 (page 1095)

What is the difference between the graphs of

G(x,y)=i+jandF(x,y)=2i+2j

Short Answer

Expert verified

The difference between the graphs are F(x,y)=2G(x,y);at(0,0).

Step by step solution

01

Step 1.Given Information

What is the difference between the graphs of

G(x,y)=i+jandF(x,y)=2i+2j

02

Step 2. The given vector is G(x,y)=i+j and F(x,y)=2i+2j.

Foreverypoint(x,y)鈭�鈩�2WecanwriteF(x,y)asF(x,y)=2(i+j)AsweknowG(x,y)=i+jSowecanwriteF(x,y)=2G(x,y)HenceF(x,y)=2G(x,y);at(0,0)

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Most popular questions from this chapter

Calculus of vector-valued functions: Calculate each of the following.

ddt(r(t)),wherer(t)=3cos2ti+5tj+tt2+1k

Find

鈭�S鈥�F(x,y,z)鈰�ndSifF(x,y,z)=2xzi+2yzj鈭�18k

and S is the portion of the hyperboloid x2+y2-9=z2that lies between the planes

z = 鈭�4 and z = 0, with n pointing outwards.

Find the flux of the given vector field through a permeable membrane described by surface S.

F(x,y,z)=yi+xj+k, where S is the paraboloid with equation role="math" localid="1650299986090" z=4x2+y2+1that lies above the annulus determined by 1鈮�x2+y2鈮�4 in the XY-plane.

F(x,y,z)=鈭�xzi鈭�yzj+z2k, where S is the cone with equation z=x2+y2between z=2,4, with n pointing outwards.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing 鈭�Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then 鈭�Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then 鈭�SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing 鈭�SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing 鈭�SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.
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